Reliability Methods for Bimodal Distribution with First-Order Approximation


In traditional reliability problems, the distribution of a basic random variable is usually unimodal; in other words, the probability density of the basic random variable has only one peak. In real applications, some basic random variables may follow bimodal distributions with two peaks in their probability density. When binomial variables are involved, traditional reliability methods, such as the first-order second moment (FOSM) method and the first-order reliability method (FORM), will not be accurate. This study investigates the accuracy of using the saddlepoint approximation (SPA) for bimodal variables and then employs SPA-based reliability methods with first-order approximation to predict the reliability. A limit-state function is at first approximated with the first-order Taylor expansion so that it becomes a linear combination of the basic random variables, some of which are bimodally distributed. The SPA is then applied to estimate the reliability. Examples show that the SPA-based reliability methods are more accurate than FOSM and FORM.


Mechanical and Aerospace Engineering

Research Center/Lab(s)

Intelligent Systems Center


Funding from the National Science Foundation (Grant No. CMMI 1562593) and the Intelligence Systems Center at Missouri University of Science and Technology.

Keywords and Phrases

Probability density function; Probability distributions; Random variables; Structural analysis, Bimodal distribution; First order reliability methods; First order second moment method; First-order approximations; Limit state functions; Probability densities; Reliability problems; Saddle-point approximation, Reliability

International Standard Serial Number (ISSN)


Document Type

Article - Journal

Document Version


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© 2019 American Society of Mechanical Engineers (ASME), All rights reserved.

Publication Date

01 Mar 2019